Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2001,2,Mod(1,2001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2001 = 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2001.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(15.9780654445\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{7} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 7\nu^{4} - 11\nu^{3} + 14\nu^{2} + 10\nu - 7 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 2\nu^{5} + 7\nu^{4} + 15\nu^{3} - 10\nu^{2} - 26\nu - 1 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 9\nu^{4} + \nu^{3} + 20\nu^{2} - 2\nu - 5 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} - 2\nu^{5} + 25\nu^{4} + 9\nu^{3} - 50\nu^{2} - 6\nu + 5 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{6} - 2\nu^{5} - 25\nu^{4} + 23\nu^{3} + 46\nu^{2} - 46\nu - 5 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + \beta_{3} + 4\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 7\beta_{5} + 5\beta_{4} - 2\beta_{3} + 6\beta_{2} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} - 10\beta_{5} - 9\beta_{4} + 8\beta_{3} - \beta_{2} + 19\beta _1 - 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} + 44\beta_{5} + 28\beta_{4} - 19\beta_{3} + 34\beta_{2} - 11\beta _1 + 81 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.37229 | 1.00000 | 3.62775 | 2.09663 | −2.37229 | −1.30993 | −3.86149 | 1.00000 | −4.97382 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.55072 | 1.00000 | 0.404722 | −0.920116 | −1.55072 | −2.53796 | 2.47382 | 1.00000 | 1.42684 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.49169 | 1.00000 | 0.225141 | −2.94997 | −1.49169 | 1.89422 | 2.64754 | 1.00000 | 4.40045 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.865893 | 1.00000 | −1.25023 | 1.66575 | −0.865893 | −0.715958 | 2.81435 | 1.00000 | −1.44236 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.17088 | 1.00000 | −0.629030 | 0.418864 | 1.17088 | 1.98148 | −3.07829 | 1.00000 | 0.490441 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.75752 | 1.00000 | 1.08886 | −2.67591 | 1.75752 | 0.0258051 | −1.60134 | 1.00000 | −4.70295 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.35219 | 1.00000 | 3.53279 | −2.63525 | 2.35219 | −4.33765 | 3.60540 | 1.00000 | −6.19860 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
| \(29\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2001.2.a.j | ✓ | 7 |
| 3.b | odd | 2 | 1 | 6003.2.a.i | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2001.2.a.j | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 6003.2.a.i | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):
|
\( T_{2}^{7} + T_{2}^{6} - 10T_{2}^{5} - 10T_{2}^{4} + 29T_{2}^{3} + 29T_{2}^{2} - 24T_{2} - 23 \)
|
|
\( T_{5}^{7} + 5T_{5}^{6} - 3T_{5}^{5} - 40T_{5}^{4} - 15T_{5}^{3} + 87T_{5}^{2} + 36T_{5} - 28 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} + \cdots - 23 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( T^{7} + 5 T^{6} + \cdots - 28 \)
|
| $7$ |
\( T^{7} + 5 T^{6} + \cdots - 1 \)
|
| $11$ |
\( T^{7} + 12 T^{6} + \cdots + 23 \)
|
| $13$ |
\( T^{7} + 13 T^{6} + \cdots - 1867 \)
|
| $17$ |
\( T^{7} + 12 T^{6} + \cdots + 12751 \)
|
| $19$ |
\( T^{7} + 5 T^{6} + \cdots - 7244 \)
|
| $23$ |
\( (T + 1)^{7} \)
|
| $29$ |
\( (T - 1)^{7} \)
|
| $31$ |
\( T^{7} + 8 T^{6} + \cdots - 8156 \)
|
| $37$ |
\( T^{7} + 24 T^{6} + \cdots + 85172 \)
|
| $41$ |
\( T^{7} - 9 T^{6} + \cdots - 1065508 \)
|
| $43$ |
\( T^{7} + T^{6} + \cdots - 15268 \)
|
| $47$ |
\( T^{7} - 27 T^{6} + \cdots - 10031 \)
|
| $53$ |
\( T^{7} + T^{6} + \cdots + 596 \)
|
| $59$ |
\( T^{7} - 8 T^{6} + \cdots + 128 \)
|
| $61$ |
\( T^{7} - T^{6} + \cdots - 36740 \)
|
| $67$ |
\( T^{7} + 16 T^{6} + \cdots + 124927 \)
|
| $71$ |
\( T^{7} + 13 T^{6} + \cdots - 16 \)
|
| $73$ |
\( T^{7} + 23 T^{6} + \cdots + 4228 \)
|
| $79$ |
\( T^{7} + 44 T^{6} + \cdots + 127580 \)
|
| $83$ |
\( T^{7} - 21 T^{6} + \cdots - 212788 \)
|
| $89$ |
\( T^{7} + 5 T^{6} + \cdots - 18865 \)
|
| $97$ |
\( T^{7} + 55 T^{6} + \cdots - 1979204 \)
|
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